Determine the length of the greater leg of a right-angled triangle, if the hypotenuse is 6√3

| January 13, 2021 | education

Determine the length of the greater leg of a right-angled triangle, if the hypotenuse is 6√3, and one of the acute angles is 30 °, find medians, heights, radii.

To solve this problem, remember that the cosine of an acute angle in a right triangle is the ratio of the adjacent leg to the hypotenuse. We calculate the length over the leg, knowing that the hypotenuse is 6√3.
cos 30 = AC / 6√3
√3 / 2 = AC / 6√3
AC = 6√3 * √3 / 2 = 6 * 3/2 = 9 cm.
The median drawn from the top of the right angle is half the hypotenuse. The radius of the circumscribed circle is equal to this median and equal to half the hypotenuse.
m = R = 6√3 / 2 = 3√3 cm.
The radius of the inscribed circle is half the sum of the legs, reduced by the hypotenuse.
We calculate the second leg by the Pythagorean theorem.
AB ^ 2 = 108 – 81 = 27
AB = 3√3 cm.
r = 3√3 + 9-6√3 / 2 = 9-3√3 / 2 cm.
Let’s calculate the height.
H = 3√3 * 9/6√3 = 27/6 = 9/2 = 4.5 cm.
Answer: 9 cm, 3√3 cm, 3√3 cm, 9-3√3 / 2 cm, 4.5 cm.



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